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https://journals.aps.org/prd/abstract/10.1103/PhysRevD.98.042006
DOI: 10.1103/PhysRevD.98.042006
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2018 by American Physical Society. All rights reserved.
DIRETORIA DE TRATAMENTO DA INFORMAÇÃO Cidade Universitária Zeferino Vaz Barão Geraldo
CEP 13083-970 – Campinas SP Fone: (19) 3521-6493 http://www.repositorio.unicamp.br
Dark Energy Survey year 1 results: Galaxy clustering for combined probes
J. Elvin-Poole,1 M. Crocce,2 A. J. Ross,3 T. Giannantonio,4,5,6 E. Rozo,7 E. S. Rykoff,8,9 S. Avila,10,11N. Banik,12,13 J. Blazek,14,15 S. L. Bridle,16R. Cawthon,17A. Drlica-Wagner,12 O. Friedrich,6,18N. Kokron,19,20 E. Krause,21 N. MacCrann,15,22J. Prat,23 C. Sánchez,23L. F. Secco,24I. Sevilla-Noarbe,10M. A. Troxel,15,22T. M. C. Abbott,25
F. B. Abdalla,26,27 S. Allam,28 J. Annis,28 J. Asorey,29,30K. Bechtol,31 M. R. Becker,21,32A. Benoit-L´evy,26,33,34 G. M. Bernstein,24E. Bertin,34,33D. Brooks,26E. Buckley-Geer,28D. L. Burke,35,21A. Carnero Rosell,36,20D. Carollo,30,37
M. Carrasco Kind,38,39 J. Carretero,23F. J. Castander,2C. E. Cunha,21C. B. D’Andrea,24 L. N. da Costa,20,36 T. M. Davis,29,30C. Davis,21 S. Desai,40H. T. Diehl,28J. P. Dietrich,41,42S. Dodelson,28,43 P. Doel,26T. F. Eifler,44,45
A. E. Evrard,46,47E. Fernandez,23B. Flaugher,28P. Fosalba,2 J. Frieman,28,43 J. García-Bellido,48 E. Gaztanaga,2 D. W. Gerdes,47,46K. Glazebrook,49D. Gruen,35,21R. A. Gruendl,38,39J. Gschwend,36,20G. Gutierrez,28W. G. Hartley,26,50 S. R. Hinton,29K. Honscheid,15,22J. K. Hoormann,29B. Jain,24D. J. James,51M. Jarvis,24T. Jeltema,52M. W. G. Johnson,38
M. D. Johnson,38A. King,29K. Kuehn,53 S. Kuhlmann,54N. Kuropatkin,28O. Lahav,26G. Lewis,55,30 T. S. Li,28 C. Lidman,53,30M. Lima,19,20 H. Lin,28E. Macaulay,29M. March,24J. L. Marshall,56P. Martini,15,57P. Melchior,58
F. Menanteau,39,38R. Miquel,23,59 J. J. Mohr,18,41,42A. Möller,60,30R. C. Nichol,61B. Nord,28C. R. O’Neill,29,30 W. J. Percival,61D. Petravick,38A. A. Plazas,45A. K. Romer,62,35M. Sako,24E. Sanchez,10V. Scarpine,28R. Schindler,35
M. Schubnell,47E. Sheldon,63M. Smith,64R. C. Smith,25 M. Soares-Santos,28F. Sobreira,20,65N. E. Sommer,60,30 E. Suchyta,66M. E. C. Swanson,38G. Tarle,47D. Thomas,61,22 B. E. Tucker,30,60D. L. Tucker,28S. A. Uddin,30,67
V. Vikram,54A. R. Walker,25R. H. Wechsler,35,32,21 J. Weller,42,18,6W. Wester,28 R. C. Wolf,24 F. Yuan,30,60B. Zhang,60,30and J. Zuntz68
(Dark Energy Survey Collaboration)
1
Jodrell Bank Center for Astrophysics, School of Physics and Astronomy, University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
2
Institute of Space Sciences, IEEC-CSIC, Campus UAB, Carrer de Can Magrans, s/n, 08193 Barcelona, Spain
3
Center for Cosmology and Astro-Particle Physics, The Ohio State University, Columbus, Ohio 43210, USA
4
Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, United Kingdom
5
Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB3 0HA, United Kingdom
6
Universitäts-Sternwarte, Fakultät für Physik, Ludwig-Maximilians Universität München, Scheinerstr. 1, 81679 München, Germany
7
Department of Physics, University of Arizona, Tucson, Arizona 85721, USA 8SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
9
Kavli Institute for Particle Astrophysics & Cosmology, P. O. Box 2450, Stanford University, Stanford, California 94305, USA
10
Centro de Investigaciones Energ´eticas, Medioambientales y Tecnológicas (CIEMAT), Madrid, Spain 11Institute of Cosmology & Gravitation, University of Portsmouth, Portsmouth, PO1 3FX, United Kingdom
12
Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, Illinois 60510, USA 13Department of Physics, University of Florida, Gainesville, Florida 32611, USA 14
Institute of Physics, Laboratory of Astrophysics, École Polytechnique F´ed´erale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix, Switzerland
15
Center for Cosmology and Astro-Particle Physics, The Ohio State University, Columbus, Ohio 43210, USA
16
Jodrell Bank Center for Astrophysics, School of Physics and Astronomy, University of Manchester, Oxford Road, Manchester, M13 9PL, United Kingdom
17
Kavli Institute for Cosmological Physics, University of Chicago, Chicago, Illinois 60637, USA 18Max Planck Institute for Extraterrestrial Physics, Giessenbachstrasse, 85748 Garching, Germany
19
Departamento de Física Matemática, Instituto de Física, Universidade de São Paulo, CP 66318, São Paulo, SP, 05314-970, Brazil
20
Laboratório Interinstitucional de e-Astronomia - LIneA, Rua Gal. Jos´e Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil
21Kavli Institute for Particle Astrophysics & Cosmology, P. O. Box 2450, Stanford University, Stanford, California 94305, USA
22Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA 23
Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona) Spain
24
Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
25
Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory, Casilla 603, La Serena, Chile
26
Department of Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT, United Kingdom
27
Department of Physics and Electronics, Rhodes University, PO Box 94, Grahamstown, 6140, South Africa
28
Fermi National Accelerator Laboratory, P. O. Box 500, Batavia, Illinois 60510, USA 29School of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072, Australia
30
ARC Centre of Excellence for All-sky Astrophysics (CAASTRO) 31LSST, 933 North Cherry Avenue, Tucson, Arizona 85721, USA 32
Department of Physics, Stanford University, 382 Via Pueblo Mall, Stanford, California 94305, USA 33CNRS, UMR 7095, Institut d’Astrophysique de Paris, F-75014, Paris, France
34
Sorbonne Universit´es, UPMC Univ Paris 06, UMR 7095, Institut d’Astrophysique de Paris, F-75014, Paris, France
35
SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA 36Observatório Nacional, Rua Gal. Jos´e Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil
37
INAF - Osservatorio Astrofisico di Torino, Pino Torinese, Italy
38National Center for Supercomputing Applications, 1205 West Clark St., Urbana, Illinois 61801, USA 39
Department of Astronomy, University of Illinois, 1002 W. Green Street, Urbana, Illinois 61801, USA 40Department of Physics, IIT Hyderabad, Kandi, Telangana 502285, India
41
Faculty of Physics, Ludwig-Maximilians-Universität, Scheinerstr. 1, 81679 Munich, Germany 42Excellence Cluster Universe, Boltzmannstr. 2, 85748 Garching, Germany
43
Kavli Institute for Cosmological Physics, University of Chicago, Chicago, Illinois 60637, USA 44Department of Physics, California Institute of Technology, Pasadena, California 91125, USA
45
Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr., Pasadena, California 91109, USA 46
Department of Astronomy, University of Michigan, Ann Arbor, Michigan 48109, USA 47Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA 48
Instituto de Fisica Teorica UAM/CSIC, Universidad Autonoma de Madrid, 28049 Madrid, Spain 49Centre for Astrophysics & Supercomputing, Swinburne University of Technology,
Victoria 3122, Australia
50Department of Physics, ETH Zurich, Wolfgang-Pauli-Strasse 16, CH-8093 Zurich, Switzerland 51
Astronomy Department, University of Washington, Box 351580, Seattle, Washington 98195, USA 52Santa Cruz Institute for Particle Physics, Santa Cruz, California 95064, USA
53
Australian Astronomical Observatory, North Ryde, NSW 2113, Australia 54Argonne National Laboratory, 9700 South Cass Avenue, Lemont, Illinois 60439, USA 55
Sydney Institute for Astronomy, School of Physics, A28, The University of Sydney, NSW 2006, Australia 56George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy, and Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843, USA
57Department of Astronomy, The Ohio State University, Columbus, Ohio 43210, USA 58
Department of Astrophysical Sciences, Princeton University, Peyton Hall, Princeton, New Jersey 08544, USA 59
Institució Catalana de Recerca i Estudis Avançats, E-08010 Barcelona, Spain 60The Research School of Astronomy and Astrophysics, Australian National University,
ACT 2601, Australia
61Institute of Cosmology & Gravitation, University of Portsmouth, Portsmouth, PO1 3FX, United Kingdom 62
Department of Physics and Astronomy, Pevensey Building, University of Sussex, Brighton, BN1 9QH, United Kingdom
63
Brookhaven National Laboratory, Bldg 510, Upton, New York 11973, USA
64School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, United Kingdom 65
Instituto de Física Gleb Wataghin, Universidade Estadual de Campinas, 13083-859, Campinas, SP, Brazil
66Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
67Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing, Jiangshu 210008, China 68
Institute for Astronomy, University of Edinburgh, Edinburgh EH9 3HJ, United Kingdom (Received 8 August 2017; published 27 August 2018; corrected 28 August 2018) We measure the clustering of DES year 1 galaxies that are intended to be combined with weak lensing samples in order to produce precise cosmological constraints from the joint analysis of large-scale structure and lensing correlations. Two-point correlation functions are measured for a sample of6.6 × 105luminous red galaxies selected using theREDMAGIC algorithm over an area of 1321 square degrees, in the redshift range0.15 < z < 0.9, split into five tomographic redshift bins. The sample has a mean redshift uncertainty ofσz=ð1 þ zÞ ¼ 0.017. We quantify and correct spurious correlations induced by spatially variable survey properties, testing their impact on the clustering measurements and covariance. We demonstrate the sample’s robustness by testing for stellar contamination, for potential biases that could arise from the systematic correction, and for the consistency between the two-point auto- and cross-correlation functions. We show that the corrections we apply have a significant impact on the resultant measurement of cosmological parameters, but that the results are robust against arbitrary choices in the correction method. We find the linear galaxy bias in each redshift bin in a fiducial cosmology to be bðσ8=0.81Þjz¼0.24¼ 1.40 0.07, bðσ8=0.81Þjz¼0.38¼ 1.60 0.05, bðσ8=0.81Þjz¼0.53¼ 1.60 0.04 for galaxies with luminosities L=L>0.5, bðσ8=0.81Þjz¼0.68¼ 1.93 0.04 for L=L>1 and bðσ8=0.81Þjz¼0.83¼ 1.98 0.07 for L=L>1.5, broadly consistent with expectations for the redshift and luminosity dependence of the bias of red galaxies. We show these measurements to be consistent with the linear bias obtained from tangential shear measurements.
DOI:10.1103/PhysRevD.98.042006
I. INTRODUCTION
Galaxies are a biased tracer of the matter density field. In the standard“halo model” paradigm, they form in collapsed over-densities (dark matter halos[1]), and the mass of the halo they reside in is known to correlate with the luminosity and color of the galaxy, with more luminous and redder galaxies strongly correlated with higher mass. Therefore, the galaxy“bias” depends strongly on the particular sample being studied. Thus, in cosmological studies the galaxy bias is often treated as a nuisance parameter—one that is degenerate with the amplitude of the clustering of matter. See, e.g., [2] and references therein.
The degeneracy can be broken with additional observ-ables. This includes the weak gravitational lensing“shear” field, which is induced by the matter density field. Correlation between the galaxies and the shear field ([3]; often referred to as“galaxy-galaxy lensing”) contains one factor of the galaxy bias and two factors of the matter field. The galaxy autocorrelation contains two factors of the galaxy bias and again two factors of the matter field. Thus, the combination of the two measurements can break the degeneracy between the two quantities, and it is a sensitive probe of the late-time matter field (see, e.g.,[4,5]).
The autocorrelation of the shear field alone includes no factors of the galaxy bias and can thus be used directly as a probe of the matter field. However, its sensitivity to many systematic uncertainties related to the estimation of the shear field differs from that of the galaxy-galaxy lensing
signal. As shown by[6–9], the impact of such systematic uncertainties can be mitigated by combining the shear autocorrelation measurements with those of galaxy cluster-ing and galaxy-galaxy lenscluster-ing. Thus there is substantial gain to be obtained from a combined analysis.
Such a combined analysis is performed with the Dark Energy Survey (DES [10]; [11]) year 1 (Y1) data ([12]; hereafter, Y1COSMO). DES is an imaging survey currently amassing data over a5000 deg2footprint in five passbands (grizY). When completed, it will have mapped 300 million galaxies and tens of thousands of galaxy clusters.
In this work, we study the clustering of red sequence galaxies selected from DES Y1 data using theREDMAGIC
[13]algorithm, chosen for its small redshift uncertainty. We study the same sample used to obtain cosmological results in the Y1COSMO combined analysis. In particular, we study the large-scale clustering amplitude and its sensitivity to observational systematics. Following previous studies
[14–17], we use angular maps to track the observing conditions of the Y1 data in order to identify and correct for spurious fluctuations in the galaxy density field. We further determine the effect the corrections have on the covariance matrix of the angular autocorrelation of the galaxies. We present robust measurements of the clustering amplitude ofREDMAGIC galaxies as a function of redshift and luminosity, thus gaining insight into the physical nature of these galaxies and how they compare to other red galaxy samples. The results of this paper are then used for the joint DES cosmological analysis presented in Y1COSMO.
This outline of this paper is: we summarize in Sec.IIthe model we use to describe our galaxy clustering measure-ments; we present in Sec.IIIthe DES data we use; Sec.IV
presents how we measure clustering statistics and estimate their covariance; Sec. V summarizes the results of our observational systematic tests. We present our primary results with galaxy bias measurements in Sec. VI and a demonstration of their robustness in Sec. VII before concluding in Sec.VIII.
In order to avoid confirmation bias, we have performed this analysis “blind”: we did not measure parameter con-straints or plot the correlation function measured from the data on the same axis as any theoretical prediction or simulated clustering measurement until the sample and all measurements in Y1COSMO were finalized.
Unless otherwise noted, we use a fiducialΛCDM cosmol-ogy, fixing cosmological parameters at Ωm ¼ 0.295,
As¼ 2.260574 × 10−9, Ωb¼ 0.0468, h ¼ 0.6881, ns¼ 0.9676. This is consistent with the latest cosmological data from the Planck mission [18] and is used as the fiducial cosmology for all the DES Y1 analyses used in Y1COSMO. We use this cosmology to generate Gaussian mocks in Sec.Vfor systematics testing.
After unblinding, we remeasure the galaxy bias, fixing the cosmological parameters at the mean of the DES Y1COSMO posterior, Ωm¼0.276, As¼2.818378×10−9, Ωb¼0.0531, h¼ 0.7506, ns¼ 0.9939, Ων¼ 0.00553 (note that we show these values at a greater precision than can be measured by DES). This cosmology was used for all bias measure-ments in Sec.VI.
II. THEORY
Throughout this paper, we modelREDMAGIC clustering
measurements assuming a local, linear galaxy bias model
[19], where the galaxy and matter density fluctuations are related byδgðxÞ ¼ bδmðxÞ, with density fluctuations defined byδ ≡ ðnðxÞ − ¯nÞ=¯n. The validity of this assumption over the scales considered here is provided in[20]and shown in simulations in[21].
The galaxy clustering model used in this paper matches that used in Y1COSMO. This model includes three neutrinos of degenerate mass.
We consider multiple galaxy redshift bins i, each characterized by aREDMAGIC galaxy redshift distribution ni
gðzÞ, normalized to unity in redshift, and a galaxy bias bi
which is assumed to be constant across the redshift bin range. Under the Limber[22] and flat-sky approximation the theoretical prediction for the galaxy correlation function wðθÞ in a given bin is,
wiðθÞ ¼ ðbiÞ2 Z dll 2πJ0ðlθÞ Z dχ ×½n i gðzðχÞÞdz=dχ2 χ2 PNL lþ 1=2 χ ; zðχÞ ; ð1Þ
where the speed of light has been set to one,χðzÞ is the comoving distance to a given redshift (in a flat universe, which is assumed throughout); J0is the Bessel function of order zero; HðzÞ is the Hubble expansion rate at redshift z; and PNLðk; zÞ is the three-dimensional matter power spectrum at redshift z and wave number k (which, in this Limber approximation, is set equal to ðl þ 1=2Þ=χ). Note that, in Eq.(1), we have assumed the bias to be constant within each bin, see Fig. 8 in [20] for a test of this assumption. Again, all assumptions and approximations mentioned here have been shown to be inconsequential in
[20,21]. To model cross-correlation between redshift bins we simply change nigðzÞ2→ nigðzÞn
j
gðzÞ and ðbiÞ2→ bibj
in Eq.(1).
Throughout this paper, we use the COSMOSIS framework
[23]to compute correlation functions, and to infer cosmo-logical parameters. The evolution of linear density fluctua-tions is obtained using the CAMB module [24] and then converted to a nonlinear matter power spectrum PNLðkÞ
using the updatedHalofit recipe [25].
The theory modeling we use assumes the Limber approximation, and it also neglects redshift space distor-tions. For the samples and redshift binning used in this paper, those effects start to become relevant at scales of θ ≳ 1 deg [26–28]. In a companion paper [20], it is explicitly shown that they have negligible impact in derived cosmological parameters given the statistical error bars of DES Y1. Concretely a theory data vector was produced with the exact (non-Limber) formula including redshift space distortions and was then analyzed with the baseline pipeline assumed here, and also in Y1COSMO. Figure 8 in
[20]shows that including or not including these contribu-tions makes negligible impact in parameters such as Ωm
and S8for a LCDM universe or w in a wCDM one. We also tested the impact of these effects on the fixed cosmology bias measurements in Sec. VI and find them to be negligible. Hence, in what follows, such terms are ignored for speed and simplicity. However, future data analyses may need to account for these effects due to improved statistical uncertainty.
We model (and marginalize over) photometric redshift bias uncertainties as an additive shiftΔziin theREDMAGIC
redshift distribution ni
RMðzÞ for each redshift bin i.
niðzÞ ¼ niRMðz − ΔziÞ ð2Þ The priors on theΔzinuisance parameters, are measured
directly using the angular cross correlation between the DES sample and a spectroscopic sample. These values are shown in TableII, and the method is described in full in
[29]. We use the same Δzi as Y1COSMO for all tests of
robustness of the parameter constraints.
We also compare the measurements of bi to the same
quantity measured by galaxy-galaxy lensing using the two-point correlation functionγt (see [30] for definition). We
use the notation bi×for this measurement. The details of this
measurement are described in [30](hereafter Y1GGL). In order to take the off-diagonal elements of the covariance matrix between the two probes into account, we produce joint constraints from wðθÞ and γtat fixed cosmology (the
mean of the Y1COSMO posterior), using different bias parameters for the two probes, and marginalizing over the same nuisance parameters as were used in the fiducial analysis of Y1COSMO (intrinsic alignments, source and lens photo–z bias, and shear calibration). To test the consistency between the two probes we use the parameter r which is,
ri¼b
i ×
bi: ð3Þ
If r≠ 1, this indicates an inconsistency between the two bias measurements and would thus suggest a breakdown of our simple linear bias model. This test informs the choice of fixing r¼ 1 in the Y1COSMO analysis.
A combination of galaxy clustering and galaxy-galaxy lensing, provides a measurement of galaxy bias andσ8only if you assume that r¼ 1. This test provides a measurement of r which informs the choice of fixing r¼ 1 in the Y1COSMO analysis. In principle, this test could also be performed by including the shear-shear correlation which also measures σ8.
III. DATA A. Y1 gold
We use data taken in the first year (Y1) of DES observations[31]. Photometry and“clean” galaxy samples were produced with this data as outlined by[32](hereafter denoted‘Y1GOLD’). The outputs of this process represent the Y1 ‘Gold’ catalog. Data were obtained over a total footprint of ∼1800 deg2; this footprint is defined by a HEALPIX [33] map at resolution Nside¼ 4096 (equivalent
to 0.74 square arcmin) and includes only pixels with minimum exposure time of at least 90 seconds in each of the g,r, i, and z-bands, a valid calibration solution, as well as additional constraints (see Y1GOLD for details). A series of veto masks, including among others masks for bright stars and the Large Magellanic Cloud, reduce the area by∼300 deg2, leaving∼1500 deg2suitable for galaxy clustering study. We explain further cuts to the angular mask in Sec. III B. All data described in this and in subsequent sections are drawn from catalogues and maps generated as part of the DES Y1 Gold sample and are fully described in Y1GOLD.
B. REDMAGIC sample
The galaxy sample we use in this work is generated by theREDMAGIC algorithm, run on DES Y1 Gold data. The REDMAGIC algorithm selects Luminous Red Galaxies
(LRGs) in such a way that photometric redshift uncertain-ties are minimized, as is described in[13]. This method is able to achieve redshift uncertainties σz=ð1 þ zÞ < 0.02 over the redshift range of interest. The REDMAGIC
algo-rithm produces a redshift prediction zRMand an uncertainty
σz which is assumed to be Gaussian. This sample was
chosen instead of other DES photometric samples because of its small redshift uncertainty, which is obtained at the expense of number density.
TheREDMAGIC algorithm makes use of an empirical
red-sequence template generated by the training of the
REDMAPPER cluster finder [34,35]. As described in [35], training of the red-sequence template requires overlapping spectroscopic redshifts, which in this work were obtained from SDSS in the Stripe 82 region [36] and the OzDES spectroscopic survey in the DES deep supernova fields[37]. For theREDMAGIC samples in this work, we make use of two separate versions of the red-sequence training. The first is based on SExtractorMAG_AUTO quantities from the Y1 coadd catalogs, as applied to REDMAPPER in [38]. The
second is based on a simultaneous multiepoch, multiband, and multiobject fit (MOF) (see Sec. 6.3 of Y1GOLD), as applied toREDMAPPER(Mcclintock et al. 2017, in prepa-ration). In general, due to the careful handling of the point-spread function (PSF) and matched multiband photometry, theMOF photometry yields lower color scatter and, hence, smaller scatter in red-sequence photo-zs. For each version of the catalog, photometric redshifts and uncertainties are primarily derived from the fit to the red-sequence template. In addition, an afterburner step is applied (as described in Sec. 3.4 of[13]) to ensure that REDMAGIC photo-zs and
errors are consistent with those derived from the associated
REDMAPPERcluster catalog [13].
As described in[13], theREDMAGIC algorithm computes
color-cuts necessary to produce a luminosity-thresholded sample of constant co-moving density. Both the luminosity threshold and desired density are independently configu-rable, but in practice higher luminosity thresholds require a lower density for good performance. We note that in[13]the co-moving density was computed with the central redshift of each galaxy (zRM). For this work, the density was computed
by sampling from a Gaussian distribution zRM σz, which
creates a more stable distribution near filter transitions. This is the only substantial change to theREDMAGIC algorithm
since the publication of[13].
We useREDMAGIC samples split into five redshift bins of widthΔz ¼ 0.15 from z ¼ 0.15 to z ¼ 0.9. We define our footprint such that the data in each redshift bin will be complete to its redshift limit across the entire footprint. To make this possible, we define samples based on a lumi-nosity threshold. Reference luminosities are computed as a function of L, computed using a Bruzual and Charlot[39]
model for a single star-formation burst at z¼ 3 [See Sec. 3.2[35]]. Naturally, increasing the luminosity thresh-old provides a higher redshift sample as well as decreasing
the comoving number density. Using a different luminosity threshold in each redshift bin allows us to maximize signal to noise while also providing a complete sample in each redshift bin. The details of these bins are given in TableI. The five redshift bins were chosen so that the width of the bins is significantly wider than the uncertainty on individual galaxy redshifts, but smaller than the difference between the maximum redshifts of the luminosity thresh-olds used.
In addition to the primaryREDMAGIC selection, we also apply a cut on the luminosity L=L<4 as this was shown for DES Science Verification to reduce the stellar contami-nation in the sample, although this is mostly superfluous for Y1 Gold. During testing, we find that the observational systematic relationships for the 0.5L sample, used for z <0.6, are minimized for the MAG_AUTO sample, with a very minor impact on photo-z performance. For L≥ 1.0, used for z >0.6, we instead find that the observation systematic relationships are minimized for theMOF sample,
and that the photo-z performance is also improved. Consequently, we useMAG_AUTO for our z < 0.6 sample andMOF for z > 0.6. See Sec.V for further discussion.
We build the area mask for the REDMAGIC samples
based on the depth information produced with the
REDMAGIC catalogs. This information is provided by the zmax quantity, which describes the highest redshift at
which a typical red galaxy of the adopted luminosity threshold (e.g., 0.5L) can be detected at 10σ in the z-band, at5σ in the r and i-bands, and at 3σ in the g-band, as described in Sec. 3.4 of[35]. The quantity zmaxvaries from
point to point in the survey due to observing conditions. Consequently, we construct a zmax map, specified on a HEALPIX map with Nside¼ 4096. In order to obtain a
uniform expected number density of galaxies across the footprint, we only use regions for which zmaxis higher than
the upper edge of the redshift bin under consideration. The footprint is defined as the regions where this condition is met in all redshift bins. Thus, we only use pixels that satisfy each of the conditions where the0.5Lsample is complete to z¼ 0.6, the 1.0Lsample is complete to z¼ 0.75, and the1.5L sample is complete to z¼ 0.9. We also restrict the analysis to the contiguous region shown in Fig.1.
An additional 1.6% of the footprint is vetoed because it has extreme observing conditions. The selection of these cuts is detailed in Sec.V.
After masking and additional cuts, we obtain a total sample of 653,691 objects distributed over an area of 1321 square degrees, as shown in Fig. 1. The average redshift uncertainty of the sample is σz=ð1 þ zÞ ¼ 0.0166. The
redshift distribution of each bin can be seen in Fig.2. The number of objects in each bin increases up to z¼ 0.6 due to the increase in volume, and decreases at higher redshift due to the increased luminosity threshold.
TABLE I. Details of the sample in each redshift bin. Lmin=L describes the minimum luminosity threshold of the sample, ngalis the number of galaxies per square degree, and Ngal is the total number of galaxies.
z range Lmin=L
ngal
(arcmin−2) Ngal Photometry 0.15 < z < 0.3 0.5 0.0134 63719 MAGAUTO 0.3 < z < 0.45 0.5 0.0344 163446 MAGAUTO 0.45 < z < 0.6 0.5 0.0511 240727 MAGAUTO 0.6 < z < 0.75 1.0 0.0303 143524 MOF 0.75 < z < 0.9 1.5 0.0089 42275 MOF
FIG. 1. Galaxy distribution of theREDMAGIC Y1 sample used in this analysis. The fluctuations represent the raw counts, without any of the corrections derived in this analysis. We have restricted the analysis to the contiguous region shown in the figure. The area is 1321 square degrees.
IV. ANALYSIS METHODS A. Clustering estimators
We measure the correlation functions wðθÞ using the Landy and Szalay estimator [40]
ˆwðθÞ ¼DD− 2DR þ RR
RR ; ð4Þ
where DD, RR and DR are the number of pairs of galaxies from the galaxy sample D and a random catalog R. This is calculated in 20 logarithmically separated bins in angle θ between 2.5 arcmin and 250 arcmin to match the analysis in Y1COSMO. We use 60 times more randoms than data. The pair-counting was done with the package tree-corr
[41]available at https://github.com/rmjarvis/TreeCorr. We also calculate wðθÞ on Gaussian random field realizations which are described in a pixelated map format. For these correlations we use a pixel-based estimator. Using the notation of[17], the correlation between two maps N1 and N2 of mean values ¯N1, ¯N2is estimated as
ˆw1;2ðθÞ ¼ XNpix i¼1 XNpix j¼i ðNi;1− ¯N1ÞðNj;2− ¯N2Þ ¯N1¯N2 Θi;j; ð5Þ
where the sum runs through all pairs of the Npixpixels in the footprint, Ni;1is the value of the N1map in pixel i, and Θi;j is unity when the pixels i and j are separated by an
angle θ within the bin size Δθ. We have tested that the difference between the estimators in Eqs. (4) and (5) is negligible for this analysis.
B. Covariances
The fiducial covariance matrix we use for the wðθÞ measurement is a theoretical halo model covariance,
described and tested by [20]. The covariance is generated usingCosmoLike[42], and is computed by calculating the four-point correlation functions for galaxy clustering in the halo model. Additionally, an empirically determined correc-tion for the survey geometry’s effect on the shot-noise component has been added. The presence of boundaries and holes decrease the effective number of galaxy pairs as a function of pair separation, which in turn raises the error budget associated to shot noise over the standard uniform sky assumptions. This same covariance is used for the combined probes analysis and is detailed in Y1COSMO.
For the analysis of observational systematics and their correlation with the data, we use a set of 1000 mock surveys (hereafter ‘mocks’) based on Gaussian random field real-izations of the projected density field. These are then used to obtain an alternative covariance, which includes all the mask effects as in the real data. The mocks we use were produced using the following method. We first calculate, using CAMB
[24], the galaxy clustering power spectrum Cggi ðlÞ, assuming the fiducial cosmology with fixed galaxy bias bi for each
redshift bin i; the galaxy bias values are listed in TableII. The angular power spectrum is then used to produce a full-sky Gaussian random field ofδg. We apply a mask to this field
corresponding to the Y1 data, as shown in Fig.1. This is converted into a galaxy number count Ngalas a function of sky position, with the same mean as the observed number count ¯Noin each redshift bin, using
Ngal ¼ ¯Noð1 þ δgÞ: ð6Þ
Shot noise is finally added to this field by Poisson sampling the Ngal field.
In order to avoid pixels withδg<−1, which cannot be Poisson sampled, we follow the method used by [20]: before Poisson sampling, we multiply the density field by a factorα, where α < 1; we then rescale the number density ngal by1=α2in order to preserve the ratio of shot-noise to
sample variance; we then rescale the measured wðθÞ by 1=α2 to obtain the unbiased wðθÞ for each mock. This
procedure is summarized by FIG. 2. Redshift distribution of the combined REDMAGIC
sample in 5 redshift bins. They are calculated by stacking Gaussian PDFs with mean equal to the REDMAGIC redshift prediction and standard deviation equal to theREDMAGIC red-shift error. Each curve is normalized so that the area of each curve matches the number of galaxies in its redshift bin.
TABLE II. Details of the fiducial parameters used for covari-ance and parameter constraints. Here, bi
fid is the fiducial linear galaxy bias for bin i applied to the Gaussian mock surveys we use to construct the covariance matrices. TheΔziprior is a Gaussian prior applied to the additive redshift bias uncertainty. These were selected to match the analysis in (DES Collaboration et al.; Y1COSMO). z range bi fid Δzi 0.15 < z < 0.3 1.45 Gauss (0.008,0.007) 0.3 < z < 0.45 1.55 Gauss (−0.005, 0.007) 0.45 < z < 0.6 1.65 Gauss (0.006,0.006) 0.6 < z < 0.75 1.8 Gauss (0.00,0.010) 0.75 < z < 0.9 2.0 Gauss (0.00,0.010)
δg→ αδg; ð7Þ
ngal→ ngal=α2; ð8Þ wðθÞ → wðθÞ=α2: ð9Þ We then use these mocks to estimate statistical errors in galaxy number density as a function of potential system-atics. Alternatively we “contaminate” each of the 1000 mocks with survey properties as discussed in Sec. V to assess the impact of systematics on the wðθÞ covariance. Note that these mocks would not be fully realistic for wðθÞ covariance and cosmological inference as they are basically Gaussian realizations. These mocks allow us to quantify significances (i.e., a Δχ2) to null tests, which are a necessary step in our analysis. Further, given such a large number of realizations we are able to obtain estimates of both the impact of the systematic correction on the resulting statistical uncertainty and any bias imparted by our meth-odology to well below 1σ significance (e.g., given 1000 mocks, a 0.1σ bias can be detected at 3σ significance).
V. SYSTEMATICS A. Survey property (SP) maps
The number density of galaxies selected based on their imaging is likely to fluctuate with the imaging quality due to fluctuations in the noise (e.g., Malmquist bias) and limitations in the reduction pipeline. Such fluctuations can imprint the structure of certain survey properties onto the galaxy field, thereby producing a noncosmological signal. In order to quantify the extent of these correlations and remove their effect from the two-point function, maps of DES imaging properties were produced using the methods described in Ref. [43]. We consider the possibility that depth, seeing, exposure time, sky brightness and airmass, in each band griz, affect the density of galaxies we select.
In total, we consider 21 survey property maps. We refer to these as SP maps from here on:
(i) depth: the magnitude limit at which we expect to be able to detect a galaxy to 10σ significance; (ii) seeing FWHM: the full width at half maximum of
the PSF of a point source;
(iii) exposure time: total exposure time in a given band; (iv) sky brightness: the brightness of the sky, e.g., due to
background light or the Moon phase;
(v) airmass: the amount of atmosphere a source has passed through, normalized to be 1 when pointing at zenith.
Where relevant, we use the weighted average quantity over all exposures contributing to a given area.
We also consider Galactic extinction and stellar con-tamination (or obscuration [14]) as potential systematics. The stellar density map was created by selecting moder-ately bright, high confidence stars. Using the notation of Y1GOLD, this selection is MODEST_CLASS ¼ 2 with 18.0<i<20.5, FLAGS_GOLD ¼ 0, and BADMASK ≤ 2. We also include an additional color cut of0.0<g−i<3.5
and g− r > 0. These stars were binned in pixels with Nside¼ 512 (equivalent to 47 square arcmin), and the corresponding area for each pixel was computed at higher resolution (Nside¼ 4096) from the Y1 Gold footprint and
pixel coverage fraction, as well as the bad region mask. Together, this yields the number of moderately bright stars per square degree that can be used to cross-correlate galaxies with stellar density. Using MODEST_CLASS to select stars means this map could potentially contain DES galaxies. For this reason, we test for correlations with the astrophysical maps separately to the SP maps. As we find no correlation between stellar density and galaxy density, we do not take this contamination into account. For Galactic extinction, we use the standard map from[44].
B. Systematic corrections
This section describes the method used to identify and correct for observational systematics. We also discuss the uncertainty on this correction and its impact on the wðθÞ covariance. Our approach is to first identify maps that are correlated with fluctuations in the galaxy density field at a given significance. We then correct for the contamination using weights to be applied to the galaxy catalog.
As demonstrated by[45], when testing a large number of maps one expects there to be some amount of covariance between the maps and the true galaxy density field due to chance. Consequently, it is possible to over-correct the galaxy density field using the type of methods employed in this work. To limit this effect, we do not correct for all possible maps, and limit ourselves to those maps that are detected to be correlated with the galaxy density field at high significance (above a given threshold). We test the robustness of the results on our choice of threshold in Sec.VII Aand we test for biases due to over-correction in Sec.VII C. The end result of our procedure is a measurement of wðθÞ that is free of systematics above a given significance (in our concrete case a galaxy density free of two sigma correlations with SP maps, as defined below, and visualized in Fig.3) and that can be directly utilized in combination with weak lensing measurements for cosmological analyses.
We identify the most significant SP maps as follows. First, given an SP map of some quantity s, we identify all pixels in some bin s∈ ½smin; smax. We then compute the
average density of galaxies in these pixels. By scanning across the whole range of possible s-values for the SP map, we can directly observe how the galaxy density field scales with s. Examples of these type of analyses are shown in Figs.3–5.
We first remove regions of the footprint that display either especially significant (>20%) changes in galaxy density from the mean, or are poorly fit by a monotonic function. These regions are defined from the cuts shown in Fig 3. We remove regions of the footprint with i-band FWHM >4.5 and i-band exposure time >500s. These cuts remove 1.6% of the Y1 area.
FIG. 3. Correlations of volume-limitedREDMAGIC galaxy number density with seeing FWHM and exposure time before any survey property (SP; see text for more details) cuts (illustrated with the red vertical lines) were applied to the mask. In the absence of systematic correlations, the results obtained from these samples are expected to be consistent with no trend (the reference green dashed line). The cuts removed regions with i-band FWHM >4.5 pixels and i-band exposure time >500s as these showed correlations that differed significantly from the mean (>20%) or were not well fit by a monotonic function. No SP weights were used in this figure.
FIG. 4. Galaxy number density divided by the mean number density across the footprint for each redshift bin, split by the number density of stars. The points with error-bars display the results for our3Δχ2ð68Þ weighted sample, the cyan curves display the results without these weights. For the weighted sample, theχ2of the line Ngal=hNgali ¼ 1 with the data points shown for each bin is 24.9, 16.0, 13.1, 6.6 and 10.9 with Nd:o:f¼ 10. The Δχ2between the null signal and a linear best fit is 0.99, 0.95, 0.24, 0.013, and 0.082. This does not meet either of the Δχ2thresholds used in this analysis. We, therefore, see no evidence for stellar contamination or obscuration in this sample.
FIG. 5. Galaxy number density in the highest redshift bin,0.75 < z < 0.9, as a function of two example SP maps, FWHM r-band and FWHM i-band. The black points correspond to the3Δχ2ð68Þ weighted sample, the cyan line is the unweighted sample. In this redshift bin, the SP maps used in the3Δχ2ð68Þ weights were Airmass i and FWHM r. The left panel demonstrates the effect of the weights on the FWHM r correlation. The right panel demonstrates that correlations with SP maps that were not included in the weights are still reduced due to correlations among the SP maps. The full set of SP correlations for the maps in Table IIIare shown in AppendixA.
After cutting the footprint, we determine which SP maps most significantly correlate with the data by fitting a linear function to each number density relationship. We minimize aχ2model where the model is Ngal ∝ As þ B. We determine
the significance of a correlation based on the difference in χ2between the best-fit linear parameters, and a null test of
Ngal=hNgali ¼ 1,
Δχ2¼ χ2
null− χ2model: ð10Þ
The Δχ2 is then compared to the same quantity measured on the Gaussian random fields described in Sec.IV B. We then label each potential systematic to be significant at1σ if the Δχ2measured on the data is larger than 68% of the mocks respectively. We denote this threshold as Δχ2ð68Þ and quote significances as Δχ2=Δχ2ð68Þ; the square-root of this number should
roughly correspond to the significance in terms of σ. Some examples of these tests for the observational
FIG. 6. The significance of each systematic correlation. The significance is calculated by comparing theΔχ2measured on the data to the distribution in the mock realizations. We find the 68th percentileΔχ2value, labeling itΔχ2ð68Þ, for each map obtained from the mock realizations. We quote the significance for the relationship obtained on the data asΔχ2=Δχ2ð68Þ. Weights are applied for the SP map with the largest significance, with the caveat that we do not correct for both depth and the components of depth (e.g., exposure time, PSF FWHM) in the same band. For example, in the bin0.15 < z < 0.3, correcting for r-band depth (the most significant contaminant) did not remove all the r-band correlations withΔχ2=Δχ2ð68Þ > 2, so is not included in the final 2Δχ2=Δχ2ð68Þ weights. This is repeated iteratively until all maps are below a threshold significance, shown here for thresholds of2Δχ2=Δχ2ð68Þ and 3Δχ2=Δχ2ð68Þ. The x axis is shown in order of decreasing significance for the unweighted sample. The labels in bold are the SP maps included in the 2Δχ2=Δχ2ð68Þ weights. In the second redshift bin, 0.3 < z < 0.45, the 3Δχ2=Δχ2ð68Þ and 2Δχ2=Δχ2ð68Þ weights are the same because correcting for only g-band depth removes all correlations withΔχ2=Δχ2ð68Þ > 2.
systematics can be seen in Fig.6. The full set of tests can be seen in AppendixA. We see no significant correlation with stellar density in the sample, as shown in Fig. 4. Similarly, we find no correlations with Galactic extinc-tion. Thus, our main tests are against SP maps, which are particular to DES observations.
Once we identify the most significant contaminant SP maps, we define weights to be applied to the galaxy sample in order to remove the dependency, following a method close to that of the latest LSS survey analysis[5,46–48]. Note however that we identify systematics using a rigorous χ2 threshold significance criteria, based on a large set of Gaussian realizations, which to our knowledge was not done before.
For this method we apply the following steps to each redshift bin separately. The correlation with a systematic s is fitted with a function Ngal=hNgali ¼ FsysðsÞ.
For depth and airmass, the function used was a linear fit in s. For exposure time and sky brightness, the function was linear inpffiffiffis, as this is how these quantities enter the depth map. For the seeing correlations, we fit the model
Ngal=hNgali ¼ FsysðsFWHMÞ
FsysðsFWHMÞ ¼ A 1 − erf sFWHM− B σ ; ð11Þ where sFWHMis the seeing full-width half-max value, and A, B
and σ are parameters to be fitted. This functional form matches that applied to BOSS[47,49]; we believe it is thus the expected form when morphological cuts are applied to reject stars (as this is what causes the relationship for BOSS). Each galaxy i in the sample is then assigned a weight 1=FsysðsiÞ where si is the value of the systematic at the
galaxy’s location on the sky. This weight is then used when calculating wðθÞ and in all further null tests.
In this sample, we find evidence of multiple systematics at a significance of Δχ2=Δχ2ð68Þ > 3, some of which are correlated with each other. To account for this, we first apply weights for the systematic with the highestΔχ2=Δχ2ð68Þ. Then, using the weighted sample, we remeasure the signifi-cance of each remaining potential systematic and repeat the process until there are no systematics with a significance greater than a Δχ2=Δχ2ð68Þ ¼ 3 threshold. The final weights are the product of the weights from each required systematic. We also produce weights using a threshold of Δχ2=Δχ2ð68Þ ¼ 2, allowing us to determine if using a
greater threshold has any impact on our clustering measure-ments. We refer to these weights as the 3Δχ2ð68Þ and 2Δχ2ð68Þ weights respectively.
The final weights used in this sample are described in TableIII. The SP maps are either the depth or properties that contribute to the depth (e.g., holding everything else fixed, a longer exposure time will result in an increased depth). Thus, in bins where multiple SP weights were required, we avoided correcting for both depth and SPs that contribute to the depth in the same band. In these cases, we weight for only the SPs that contribute to the depth. Fig. 13shows the correlation
between the sample density and the SP maps used in TableIII, both with and without weights.
Fig.6summarizes the results of our search for contami-nating SPs, for each redshift bin. The blue points show the significance for each map, prior to the application of any weights. The black and red points display the significance after applying the3Δχ2ð68Þ and 2Δχ2ð68Þ weights respec-tively. In Sec.VII, we will test our results with both choice of weights and whether to expect any bias from over-correction from either choice.
When FsysðsÞ is a linear function, the method described
above, hereby referred to as the weights method, should be equivalent to the method used in[15,17]. This has been shown in[5]for the DES science verification redMaGiC sample.
The impact of the SP weights on the wðθÞ measurement can be seen in Fig.7. The dashed line displays the measurement with no weights applied. One can see that in all redshift bins, the application of the SP weights reduces the clustering amplitude and that the effect is greatest on large scales. This is consistent with expectations (see, e.g., Ref.[14]).
VI. RESULTS: GALAXY BIAS AND STOCHASTICITY
In this section, we present measurements of galaxy bias bi and stochastic bias ri. The amplitude of the galaxy
clustering signal is determined by the combination of TABLE III. List of the maps used in the SP weights. Each of these has been determined to impart fluctuations in our galaxy sample at >3Δχ2ð68Þ or > 2Δχ2ð68Þ significance. The weights were applied serially for each map in the order shown, starting from the top of the table.‘FWHM’ refers to the full-width-half-maximum size of the PSF. The photometric band of each SP map is in parentheses. z range Maps included in3Δχ2ð68Þ weights Maps included in2Δχ2ð68Þ weights 0.15 < z < 0.3 Depth (r) Exptime (i)
FHWM (z) FWHM (r) Airmass (z) 0.3 < z < 0.45 Depth (g) Depth (g) 0.45 < z < 0.6 FWHM (z) FWHM (z) Exptime (g) Exptime (g) FWHM (r) FWHM (r) Skybright (z) Skybright (z) Depth (i) 0.6 < z < 0.75 FWHM (gri) PCA-0 FWHM (gri) PCA-0
Skybright (r) Skybright (r) FWHM (z) FWHM (z)
Exptime (i) Exptime (z) 0.75 < z < 0.9 Airmass (i) Airmass (i) FWHM (r) FWHM (r) FWHM (g)
parametersðbiσ
8Þ2. Equivalently the galaxy-galaxy lensing
signal γt is sensitive to bi×ðσ8Þ2. In the Y1COSMO
combined probes analysis, cosmic shear provides a meas-urement ofσ8 meaning that galaxy clustering and galaxy lensing can each provide an independant measurment of galaxy bias (and, therefore, one could measure r). In this analysis, we fixσ8at the mean of the Y1COSMO postierior (σ8¼ 0.81) to measure biðσ8=0.81Þ from clustering and ri from γt. This provides a cosmology dependent
measure-ment of bias from clustering alone, and test of the assumption r¼ 1 in Y1COSMO.
The wðθÞ autocorrelation functions of the REDMAGIC
galaxy sample are shown in Fig.7. We show the autocorre-lation calculated with and without a correction for observa-tional systematics, as described in SecV. A minimum angular scaleθi
minhas been applied to each redshift bin i. These were
chosen to beθ1min¼ 430,θ2min¼ 270,θ3min¼ 200,θ4min¼ 160, andθ5min¼ 140to match the analysis in Y1COSMO. These minimum angular scales, varying with redshift, correspond to a single minimum co-moving scale R¼ 8 Mpc h−1such that θi
min¼ R=χðhziiÞ, where hzii is the mean redshift of
galaxies in bin i [20]. The scale was chosen so that a significant nonlinear galaxy bias or baryonic feedback component to the Y1COSMO data vector would not bias the cosmological parameter constraints.
The angular correlation function has been calculated on scales belowθi
min, but these were removed in all parameter
constraints.
Fixing all cosmological parameters, includingΩm, at the
Y1COSMO values, we measure the linear bias to be
b1¼ 1.40 0.07, b2¼ 1.60 0.05, b3¼ 1.60 0.04, b4¼ 1.93 0.04, and b5¼ 1.98 0.07. These bias values are shown in TableIV. Theχ2values of the combined fit and the individual bins are shown in TableV. We note that the bin with the smallest probability is bin 1.
The combined goodness-of-fit χ2 of the bias measure-ments isχ2¼ 67 and the number of degrees of freedom is ν ¼ 54 − 10 (the 10 parameters are bi,Δzi). These values
provide a probability to exceed of 1.4%. As in Y1COSMO, we note that the formal probabilities of aχ2distribution are not strictly applicable in this case due to the uncertainty on the estimates of the covariance. Further, because the fiveΔzi
are nuisance parameters with tight priors, we also consider ν ¼ 49, which yields a probability to exceed of 4.5%. These probabilities are very similar to the values obtained by the full Y1COSMO data vector, of which this is one part.
We also note that theχ2is sensitive to the inclusion of the shot-noise correction applied to the covariance detailed in FIG. 7. Two-point correlation functions for the fiducial analysis in each of the 5 redshift bins. These panels show the autocorrelation used in Y1COSMO and the galaxy bias measurements presented in this work. A correction for correlations with survey properties is applied according to the methodology in Sec.V. The grey dashed line is the correlation function calculated without the SP weights. The black points use the2Δχ2ð68Þ weights. We show correlations down to θ ¼ 2.50to highlight the goodness of the fit towards small scales, but data points within grey shaded regions have not been used in bias constraints or the galaxy clustering part of Y1COSMO. That scale cut has been set in co-moving coordinates at8 Mpc h−1. The solid red curve is the best-fit model using only the wðθÞ autocorrelations at fixed cosmology, usingΔzi priors from[29]. The solid blue curve is the best-fit model from the full cosmological analysis in Y1COSMO.
TABLE IV. The measurements of galaxy bias biand the ratio of bias from clustering and galaxy-galaxy lensing ri for each redshift bin i, calculated with cosmological parameters fixed at the mean of the Y1COSMO posterior, varying only bias and nuisance parameters with lens photo–z priors from[29]. z range biðσ 8=0.81Þ ri 0.15 < z < 0.3 1.40 0.072 1.10 0.08 0.3 < z < 0.45 1.60 0.051 0.97 0.06 0.45 < z < 0.6 1.60 0.039 0.91 0.08 0.6 < z < 0.75 1.93 0.045 1.02 0.13 0.75 < z < 0.9 1.98 0.070 0.85 0.28
Y1COSMO whereas the bi values and uncertainty were
insensitive to this change.
For the L=L>0.5 sample, the bias is nearly constant as a function of redshift, though there is a decrease at low redshift that has more than 2σ significance (the correlation in the
measured bias for bins 1 and 3 is only -0.04, so we can safely ignore it in this discussion). The difference between bin 1 and bin 3 is less significant if we determine the expectation for a passively evolving sample as in[50,51], which predicts a bias of 1.52 at z¼ 0.24 given a bias of 1.61 at z ¼ 0.53. The bias increases for the higher luminosity sample, as expected. The results are broadly consistent with previous studies of the bias of red galaxies at low redshift (see, e.g.,[52]for a review) and BOSS at intermediate redshifts (see, e.g.,[53]). Further study of the details of the REDMAGIC samples is warranted, especially if one wishes to use wðθÞ at scales smaller than those studied in Y1COSMO.
We compare these bias constraints to those measured from the galaxy-galaxy lensing probe of the same
REDMAGIC sample, presented in Y1GGL. We parameterize
the difference between the two measurements with the cross-correlation coefficients ri, which are presented in
Fig.8. Beyond linear galaxy bias, r can deviate from 1 and acquire scale dependences, and it must be properly mod-eled to constrain cosmology with combined galaxy cluster-ing and galaxy-galaxy lenscluster-ing (e.g.,[54]). We constrain ri at fixed cosmology using the Y1COSMO covariance, which includes the covariance between the two probes. TABLE V. Theχ2, and probability of obtaining aχ2exceeding
this values for each redshift bin and for all bins combined. For the combinedχ2, the number of free parameters is 10 (5 bi and 5 Δzi). The individual z binχ2values are calculated using the best fit to all z bins combined. The covariance between z bins is sufficiently small that we can treat these as independent. We have, therefore, considered each individual bin to have 2 free param-eters. It is expected that measuring the bias in each bin separately would have resulted in a smallerχ2.
z range χ2 Ndata prob
0.15 < z < 0.3 14.8 8 2.2% 0.3 < z < 0.45 6.9 10 55% 0.45 < z < 0.6 17.7 11 3.9% 0.6 < z < 0.75 11.0 12 35.9% 0.75 < z < 0.9 16.5 13 12.2% wðθÞ all bins 67.2 54 1.4%
FIG. 8. Constraints on the ratio, r, of galaxy bias measured on wðθÞ and measured from the galaxy-galaxy lensing signal (see[30] denoted Y1GGL in the text) in each redshift bin. The histograms show the posterior distributions of rifrom an MCMC fit for each z in i. The bottom-right panel displays the individual measurements for each bin (purple for our wðθÞ measurements and orange for those obtained in Y1GGL). All cosmological parameters were fixed at the DES Y1COSMO posterior mean values, and all nuisance parameters were varied as in Y1COSMO. The constraints were calculated using the full Y1COSMO covariance matrix, so the covariance between the two probes has been taken into account. We see no significant evidence for r≠ 1 within the errors.
All the nuisance parameters discussed in Y1COSMO are varied for this constraint. With our choice of scale cuts, we see no evidence of tension between the two bias measure-ments. This provides further justification for fixing r¼ 1 in the Y1COSMO analysis.
VII. DEMONSTRATION OF ROBUSTNESS We apply a number of null tests to our weighted sample to demonstrate its robustness. We do so by obtaining constraints on the galaxy bias and Ωm. These parameters are sensitive to both multiplicative and additive shifts in the amplitude of wðθÞ and we, therefore, believe they should encapsulate any potential systematic bias that could affect the cosmological analysis of Y1COSMO. We thus perform joint fits to the data in each redshift bin to obtain constraints on the five biandΩ
m. For these fits, we marginalize over an
additive redshift bias uncertainty described in TableII. All other cosmological parameters are fixed at the Y1COSMO cosmology and as such, this should not be interpreted as a measurement ofΩm to be used in further analyses. Results are obtained using the analysis pipeline described in[20]. We describe how wðθÞ is altered to perform each test throughout the rest of this section.
A. Selection of threshold
We test two thresholds used to determine when to apply weights based on a given SP map:3Δχ2ð68Þ and a more restrictive (i.e., more maps weighted for)2Δχ2ð68Þ. After reaching a certain threshold, we expect that the only effect from adding extra weights would be to bias the measure-ments (from over-correction) and add greater uncertainty. We test for those effects in the following subsections. Here, in order to demonstrate that our results are insensitive to the choice in threshold, the change in the measured biandΩm
must be negligible compared to its uncertainty.
Figure9shows the difference between the3Δχ2ð68Þ and 2Δχ2ð68Þ SP weights. Because the weights correction can
only decrease the wðθÞ signal, applying a stricter threshold significance is expected to move the contours towards smaller values of bi. Figure9shows that this impact is very
small compared to the overall Y1 uncertainty and we can conclude that the choice between3Δχ2ð68Þ and 2Δχ2ð68Þ weights will have negligible impact on the Y1COSMO parameter constraints (The final weights used in Y1COSMO are the2Δχ2ð68Þ weights).
Figure 9 also shows the impact of not including SP weights on the parameter constraints. Ignoring the SP correlations would have resulted in significantly biased
FIG. 9. Parameter constraints showing the impact of the SP weights, varying Ωm, 5 linear bias parameters bi, and 5 nuisance parametersΔzi. Contours are drawn at 68% and 95% confidence level. These constraints use the sameΔzipriors as Y1COSMO. The blue contour shows the constraints on wðθÞ calculated with no SP weights. The gray and red contours use SP weights removing all 2Δχ2ð68Þ and 3Δχ2ð68Þ correlations respectively. In this parameter space, ignoring the correlations with survey properties would have significantly biased the constraints from wðθÞ. As expected, the best fit when using the 2Δχ2ð68Þ weights is at smaller values of bithan the3Δχ2ð68Þ weights, although the difference is not significant compared to the size of the contour.
constraints on biandΩm. In every redshift bin, the shift is
greater than 2σ along the major axis of the ellipses. B. Estimator bias
We also test for potential bias in wðθÞ induced by over-correcting with the weights method and from correlations between the SP maps. This was done using the Gaussian mocks described in Sec.IV Busing the following method. After the galaxy over-density field has been generated in each realization, we insert the systematic correlation using FsysðsÞ
and the best-fit parameters for each of the systematics in Table III at 2Δχ2ð68Þ significance. This is equivalent to dividing each mock galaxy map by a map of the SP weights. We then produced a galaxy number count as before, also adding shot noise. We fit the parameters of FsysðsÞ to each realization and apply weights to the maps using the same method that is applied to the data. We measure wðθÞ using the pixel estimator in Eq. (5) on mocks with systematic con-tamination and correction, wweights, and on mocks with no
systematics added, wno sys. We define the bias in wðθÞ to be, west bias¼ 1 N XN i¼1 wno sys;i− XN j¼1 wweights;j ð12Þ where N is the total number of realizations. We then add west biasto the measured wðθÞ and measure biandΩm. This is
designed to test for any bias in wðθÞ induced by the estimator when using weights.
This result can be seen in Fig. 10, where it shows negligible impact on the parameter constraints.
C. False correlations
Given the large number of SP maps being used in the systematics tests, it is possible that chance correlations will appear significant and weights will be applied where no contamination has occurred, biasing the measured signal. To test this, we use the same Gaussian mocks as in Sec.VII B
with no added systematic contaminations. We measure the correlation of each mock with each of the 21 SP maps in Sec. VA, identifying any correlations above a 2Δχ2ð68Þ threshold significance.
The false correction bias wfalse bias, is then defined as the
average difference between the wðθÞ measured with no corrections, and the wðθÞ measured correcting for all correlations above the threshold using the weights method. We then add wfalse bias to the measured wðθÞ and test the impact on bi andΩ
m constraints.
This test is designed to test for any bias in wðθÞ induced by falsely correcting for SP maps that were only correlated with the galaxy density by chance.
This result is shown in Fig. 10, where wfalse bias for the
2Δχ2ð68Þ SP maps has been used. This shows a negligible
impact on the constraints. The wfalse bias for the3Δχ2ð68Þ
FIG. 10. Parameter constraints showing the impact of the estimator bias, westand false correction bias wfalse. The fiducial data vector and was calculated using the2Δχ2ð68Þ weights on the data. The westand wfalsewere measured on Gaussian mock surveys using a2Δχ2ð68Þ threshold significance. We see no evidence for significant bias in the bi,Ω
SP maps is not shown as it has an even smaller impact. This demonstrates that selecting a2Δχ2ð68Þ threshold does not induce a bias in the inferred bias parameters for the set of SP maps used in this analysis.
D. Impact on covariance
Correcting for multiple systematic correlations can alter the covariance of the wðθÞ measurement in various ways. We expect that scatter in the best-fit parameters should increase the variance, while the removal of some clustering modes should decrease it. We test the significance of any changes to the amplitude and structure of the covariance matrix using the Gaussian mocks.
For this test we use the same mocks as in Sec.VII Bwhich are ‘contaminated’ with the same systematic correlations found in the data. We fit the FsysðsÞ function to each mock and correct using weights. We then measure the correlation function wweights and calculate the covariance matrix of this
measurement. We also measure the correlation on mocks with no systematics added, wno sys, and calculate the covariance
matrix from each measurement. We calculate the galaxy bias biandΩmconstraints for each covariance matrix and test if the
resulting contours are significantly different. This test deter-mines whether this additional uncertainty needs to be con-sidered in the Y1COSMO analysis by marginalizing over the fitted parameters.
The results of this test are shown in Fig.11. We show that for the SP maps selected in this analysis, the impact on the size of the contours is negligible. We have, therefore, not included any additional parameters in the MCMC analysis to account for the uncertainty in the correction.
VIII. CONCLUSIONS
We have presented the 2-point angular galaxy correlation functions, wðθÞ, for a sample of luminous red galaxies in DES Y1 data, selected by theREDMAGIC algorithm. This
yielded a sample with small redshift uncertainty, a wide redshift range, and wide angular area. We split this sample into five redshift bins and analyzed its clustering. Our findings can be summarized as follows:
(i) We find that multiple systematic dependencies between REDMAGIC galaxy density and survey properties must be corrected for in order to obtain unbiased clustering measurements. We correct for these dependencies by adding weights to the gal-axies, following[46,47].
(ii) We demonstrate both that our methods sufficiently remove systematic contamination (no significant differences are found between applying a2Δχ2ð68Þ and3Δχ2ð68Þ threshold; see Fig.9) and that any bias resulting from our method removing true clustering modes is insignificant (see Fig. 10). We further FIG. 11. Parameter constraints showing the impact of the systematics correction on the covariance. Both contours use the fiducial theory data vector. The blue contour uses the covariance from mock surveys with no contamination added (labeled“cov: no sys”). The gray contour uses the covariance determined from mock surveys with the2Δχ2ð68Þ contaminations added (labeled “cov: 2σ sys”). These constraints use the sameΔzi priors as Y1COSMO.
demonstrate that our weighting method imparts neg-ligible changes to the covariance matrix (see Fig.11). (iii) We find the redshift and luminosity dependence of the bias of REDMAGIC galaxies to be broadly consistent with expectations for red galaxies. (iv) We find that the large-scale galaxy bias is consistent
with that determined by the Y1GGL galaxy-galaxy lensing measurements. This is consistent with r¼ 1 at linear scales, in agreement with basic galaxy formation theory, and a key assumption in the Y1COSMO analysis. (See Fig.8.)
(v) Our results give an unbiased wðθÞ data vector to be provided to the Y1COSMO analysis, and other DES year 1 combined probes analyses.
The methods we have presented, both correcting for systematic dependencies and ensuring the robustness of these corrections, can be used as a guide for future analyses. Possible improvements to the work include incorporating image simulations[55]and using mode projection techniques[16]. Our galaxy bias results can be extended to study luminosity dependence within redshift bins and to use smaller scale clustering in order to determine the HOD of
REDMAGIC galaxies. Already, our bias measurements can be used to inform simulations (e.g., for the support of DES Y3 analyses) and additional HOD information would be of further benefit.
Finally, the results presented here have been optimized for combination with other cosmological probes in Y1COSMO and our work has ensured the galaxy clustering measurements do not bias the Y1COSMO results. The analysis followed a strict blinding procedure and has yielded cosmological constraints when combined with the other 2-point functions.
ACKNOWLEDGMENTS
Figures 9 to 11 in this paper were produced with chainconsumer[56]. Funding for the DES Projects has been provided by the U.S. Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois
FIG. 12. Maps of potential sources of systematics. Shown here for i-band only. Maps in other bands show fluctuations on similar scales. Each SP map is shown at Nside¼ 1024. The stellar density map is shown at Nside¼ 512.
at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago, the Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University, Financiadora de Estudos e Projetos, Fundação Carlos Chagas Filho de Amparo `a Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Científico e Tecnológico and the Minist´erio da Ciência, Tecnologia e Inovação, the Deutsche Forschungsgemeinschaft and the Collaborating Institutions in the Dark Energy Survey. The Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investi-gaciones Energ´eticas, Medioambientales y Tecnológicas-Madrid, the University of Chicago, University College London, the DES-Brazil Consortium, the University of Edinburgh, the Eidgenössische Technische Hochschule (ETH) Zürich, Fermi National Accelerator Laboratory, the University of Illinois at Urbana-Champaign, the Institut de Ci`encies de l’Espai (IEEC/CSIC), the Institut de Física d’Altes Energies, Lawrence Berkeley National Laboratory,
the Ludwig-Maximilians Universität München and the asso-ciated Excellence Cluster Universe, the University of Michigan, the National Optical Astronomy Observatory, the University of Nottingham, The Ohio State University, the University of Pennsylvania, the University of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the University of Sussex, Texas A&M University, and the OzDES Membership Consortium. Work is based, in part, on observations at Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. The DES data management system is supported by the National Science Foundation under Grants No. AST-1138766 and No. AST-1536171. The DES partic-ipants from Spanish institutions are partially supported by MINECO under Grants No. AYA2015-71825, No. ESP2015-88861, No. FPA2015-68048, No. 2012-0234, No. SEV-2016-0597, and No. MDM-2015-0509, some of which include ERDF funds from the European Union. I. F. A. E. is partially funded by the CERCA program of the Generalitat
FIG. 13. Galaxy number density as a function of different SP maps. We show here only the correlations with SP maps used in the 2Δχ2ð68Þ weights calculation. The cyan line is the correlation of the sample without weights. The black points show the correlation after correction with the2Δχ2ð68Þ weights. The error bars were calculated by measuring the same correlation on the Gaussian mock surveys described in Sec.IV B. The significance of these correlations are shown in Fig.6.
